Hartree-Fock exchange

The purpose of this section is to explain how to compute hybrid functionals (or Hartree-Fock exchange, HFX) with CP2K in condensed phase systems. It is based on the developments described in 10.1021/ct900494g and 10.1063/1.2931945, and its efficient extension (ADMM) described in 10.1021/ct1002225.

Hartree-Fock exchange in CP2K is based on four center integrals, these are computed with an external library (libint). To do these exercises, CP2K must be linked to this library.

This approach has a computational cost that depends strongly on the nature of the basis, unless combined with ADMM (see below), do not use MOLOPT basis sets with HFX. We use basis sets from HFX_BASIS, which are suitable.

Truncated Coulomb operator

To enable HFX in the condensed phase (described at the Gamma point only), CP2K employs a truncated Coulomb operator for the exchange part. The physical picture is that we do not want to have 'self-exchange interactions' of an electron with its image in neighboring unit cells. As a rule of thumb, the maximum range (truncation radius) is L/2 where L is the smallest edge of the unit cell. The convergence of the exchange energy is exponential wrt. this radius. Typically, 5-6A provides good results, but this depends on the nature of the system, i.e. the band gap or the range of the maximally localized Wannier orbitals.

1st task : GGA restart wfn

Using the water input from the previous exercise, we will perform a single point GGA calculation to generate an initial wavefunction (wfn) restart. HFX calculations benefit from this.

Change the input to:

RUN_TYPE ENERGY

IOLEVEL MEDIUM

RESTART ON

comment section &EXT_RESTART

Run the input and rename the generated wfn file (WATER-RESTART.wfn) to WATER-RESTART-GGA.wfn.
Also make a note of the HOMO - LUMO gap [eV]

2nd task: PBE0-D3 water

To do a hybrid calculation, we just change the &XC section. Various examples can be found in the regtests, but here we employ a section equivalent to PBE0-D3.

Question: What is the HOMO-LUMO gap for this configuration ? How does this compare to the GGA result ? Adjust the fraction of exchange (modify the input in two places!) to 20% and/or 30%, how does this influence the gap ?

Truncated Coulomb operator with long range correction

Like in the HSE functional, the difference between the operator used for exchange and 1/r, can be accounted for by a special GGA exchange functional. Also for the truncated coulomb operator this is possible, and allows for xc functionals that embed very short range exchange operators only. This can be used to speedup the calculation, while retaining the benefits of HFX. The functional employed in this way smoothly goes from PBE to PBE0 as the range goes from 0 to Infinity.

3rd task

Add to the &XC_FUNCTIONAL section (i.e. in addition to &PBE) the following section:

&PBE_HOLE_T_C_LR
CUTOFF_RADIUS 2.5
SCALE_X 0.25
&END

and employ the same CUTOFF_RADIUS for the INTERACTION_POTENTIAL.

Rerun the single point energy calculation and note the band gap.

Is such a short range sufficient to have a sizable effect on the band gap ?

is HFX_MEM_INFO| Number of cart. primitive ERI's calculated very different for calculations with 2.5 and 6.0A truncation radius ?

Auxiliary Density Matrix Methods (ADMM)

ADMM is an approach to mitigate the cost of HFX for large basis sets. In particular, if MOLOPT basis sets are used, standard HFX becomes too expensive (CP2K can not deal efficiently with highly contracted AOs). In ADMM, an AUX_FIT_BASIS_SET is introduced, which is used to create an auxiliary density matrix (ADM) by projection. HFX is evaluated for this ADM, while the error introduced by using an ADM is corrected for with a GGA exchange functional.

4rd task : introduce ADMM

Make the following changes:

insert and additional line BASIS_SET_FILE_NAME BASIS_ADMM (and copy that file from cp2k/data as needed).

insert for each &KIND a line AUX_FIT_BASIS_SET cFIT3

insert a secion &AUXILIARY_DENSITY_MATRIX_METHOD

! use ADMM
&AUXILIARY_DENSITY_MATRIX_METHOD
! recommended, i.e. use a smaller basis for HFX
! each kind will need an AUX_FIT_BASIS_SET.
METHOD BASIS_PROJECTION
! recommended, this method is stable and allows for MD.
! can be expensive for large systems
ADMM_PURIFICATION_METHOD MO_DIAG
&END

In this tutorial we combine ADMM using a very small basis set (cFIT3) with a small primary basis (DZVP-GTH), so gains are small at best (and the results not very accurate). ADMM is most useful with good quality primary basis sets, such as e.g. MOLOPTs

Chasing charge localization in liquid water

The combination of truncated exchange and ADMM results in the most effective way to run AIMD with hybrid functionals. In some systems the difference between GGADFT and hybrids is very large. One such systems is liquid water after ionization (i.e. charge +1), where only with hybrids the expected species (OH radicals) are formed. See 10.1063/1.3664746.

5th task : ionized water

This task is optional, and can be done near the end if time is available.

adapt the admm input for water to reflect the ionized state:

! Charge and multiplicity
LSD
CHARGE 1
MULTIPLICITY 2

because the system is electronically very difficult initially, we'll reduce the convergence threshold EPS_SCF 1.0E-5 (twice).

Note that the WFN_RESTART_FILE_NAME must point to a GGA calculation of the same charge and multiplicity (do this calculations first).

Run single point energy calculations varying the fraction of exchange from 0.25 to 0.50, does the mulliken spin population reproduce Fig. 2 in 10.1021/ct1002225 ?

For the fraction 0.5, run AIMD for about 50-100fs (if time permits), what happens with the water molecule on which the spin was localized ? Do you results agree with 10.1063/1.3664746 ?

Required files

No new files are required for this exercise. If you're stuck, you can use the following worked out examples.